Extension card · Linguistic
2-tuple linguistic CODAS
This is the form of CODAS for situations where expert scores are chosen from a pre-declared term set, and the two distances to the negative-ideal are computed carrying the symbolic translation value rather than rounding to a single term.
Base method
CODAS →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Linguistic →
What this data type is, when to use it, how to write it in a cell: the family's full account is here.
What Changes from the Base Method?
Four things change. The two-distance comparison logic does not.
Cells. In crisp CODAS every cell is a single number. Here every cell consists of a term and a translation value. The expert's score is chosen from a nine-step term set: very low, low, fairly low, slightly low, medium, slightly high, fairly high, high, very high. The method converts every term, together with its translation value, into a single number; this conversion is one-to-one and drops no information. Criterion weights here are crisp numbers, not terms; the method does not generate weights, it takes them from outside.
Scale equalisation. Crisp CODAS divides the value by its column's largest value on a benefit criterion, and takes the ratio of the column's smallest value to the value on a cost criterion. 2-tuple linguistic CODAS does not perform this division. The term-to-number conversion already sits between 0 and 8, and no further division by magnitude is applied to these numbers. The negative-ideal point is built directly from these numbers according to each criterion's direction: the smallest number on a benefit criterion, the largest on a cost criterion, becomes the negative-ideal.
Distance. Euclidean and Taxicab distance are computed over these term-to-number converted values, in exactly the same form as in crisp CODAS: the difference is taken and multiplied by the weight. In crisp CODAS this was the difference of two numbers; here it is the difference between two term-translation pairs once reduced to numbers. Uncertainty here is not a three-cornered interval; it is a single term and the fine translation value around it.
Assessment score. The pairwise comparison rule (threshold ψ, Euclidean first, then Taxicab if the difference does not exceed the threshold) and the score itself are built in exactly the same form as in crisp CODAS. DecisionMind fixes, for classical L2T-CODAS, the nine-step term set, the term-to-number conversion, and this threshold rule. Weights are taken from outside as crisp numbers.
How to Read the Output
The assessment score is read as in crisp CODAS: it is a relative measure of position, not a percentage, and the negative-ideal is rebuilt whenever the alternative set changes. The difference lies here: the uncertainty beneath the score is not a wide triangle but a single term and the fine translation value around it. The robustness of the gap between two alternatives' scores is therefore tested by asking how many steps of term change, or which weight distribution, it rests on.
Thus instead of writing:
"2-tuple linguistic CODAS loses no information, so the result is exact"
the report should read:
"Expert scores were chosen from the term set and preserved with a translation value; the ranking is sensitive to this weight distribution, or to a term change of this many steps"
Preserving the translation value does not make the result exact; it only recovers the information that rounding to a term would have dropped.
When to Prefer This over the Base Method
This extension is used when experts assess a criterion not with a number but with a word chosen from a pre-declared term set. It is also used when there is a possibility of alternatives so close that a straight-line distance cannot distinguish between them. Without the second condition, that is, if distinguishability is not a problem, 2-tuple linguistic TOPSIS may suffice; what CODAS adds is the ability to separate close rivals using a second distance measure.
It should not be confused with the other member of the same family. If the expert gives not a single term but several terms with probabilities, probabilistic linguistic CODAS is used; here there is a single term and its translation value. The condition for staying with the base method is the same: if a criterion is measured, it stays measured and is not converted into a word. If the table is mixed, DecisionMind requires a single data type. The exit point is the same as for crisp CODAS: if no compromise is acceptable on one criterion, this extension too is compensatory and will not screen out anything below a threshold.
Mistakes Specific to This Extension
Discarding the translation value and rounding to the nearest term. Performing the term-to-number conversion without the translation value voids the method's one contribution, preventing information loss. In the illustrative example below, this does not change the ranking, but it hides how robust the gap really is.
Applying crisp CODAS's normalisation. Additionally dividing the term-to-number converted values by the column maximum needlessly shifts the negative-ideal point and undoes the directness of the term-to-number conversion.
Leaving the threshold value (ψ) unquestioned. The same mistake as in crisp CODAS applies here too: a report should not be written without testing whether the ranking between two close alternatives is sensitive to the threshold.
Changing the term set during the analysis. The number of steps and the order of the terms must remain the same for every expert and every alternative.
The governing principle is this:
2-tuple linguistic CODAS exists to carry the translation between terms through to the negative-ideal without losing information; any application that discards the translation value, adds crisp CODAS's normalisation, or leaves the threshold unquestioned erases the extension's one contribution.
Cases
The first case is DecisionMind's validation example. In the manifest, this 3×3 table is recorded as a synthetic fixture without a page reference; the figures were independently reproduced by running the DecisionMind engine directly. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's validation example): Choosing among three database providers
A university library will take out an annual subscription with one of three electronic database providers. The committee scores every provider on three criteria using a word chosen from the term set. The criteria are: content coverage, interface usability, and access-outage risk; the last is a lower-is-better criterion.
| Provider | Content coverage | Interface usability | Access-outage risk |
|---|---|---|---|
| A1 | fairly high, marked negative translation | medium | slightly high, slight negative translation |
| A2 | fairly high, marked positive translation | slightly high, slight negative translation | slightly low, slight positive translation |
| A3 | slightly high, slight negative translation | fairly high, marked negative translation | medium |
| Direction | higher is better | higher is better | lower is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method converts every term, together with its translation value, into a single number. From these numbers it builds the negative-ideal according to each criterion's direction. It then computes the weighted Euclidean and Taxicab distance and produces the assessment score through pairwise comparisons.
| Provider | Assessment score | Rank |
|---|---|---|
| A2 | 2.7745 | 1 |
| A3 | 0.0552 | 2 |
| A1 | -2.8297 | 3 |
The result reads as follows. A2 sits at the same step as A1 on content coverage but with the translation running in the opposite direction, that is, slightly higher; on access-outage risk it sits at the lowest step. Together these two carry A2 clearly ahead. A3 sits at the highest step on interface usability, but this criterion carries the lowest weight; A1, despite sitting at the same step as A2 on content coverage, remains at the highest step on access risk and falls to last place.
The committee's hesitation is this: if the weight on content coverage is lowered from 0.40 to 0.185 and the weight on interface usability is raised from 0.35 to 0.565 (access risk held at 0.25), A3 overtakes A2. The same computation then places A3 first at 1.8882 and A2 second at 1.8616; the gap between them is only 0.0266. This shows that A2's first place rests on the high weight given to content coverage, and that this first place can be lost once that weight is markedly reduced.
In the report: "With the given weights (content coverage 0.40, interface usability 0.35, access risk 0.25), A2 has the highest assessment score (2.7745). If the content-coverage weight is markedly lowered and the interface weight raised, A3 moves ahead; A2's first place is therefore sensitive to the weight distribution."
Source: DecisionMind's L2T-CODAS validation example. The term set and the library scenario were built for this card; the figures are taken from the manifest's synthetic fixture. DecisionMind's engine (src/engine) was run independently with the same manifest and kernel and matched the manifest's expected values exactly (A1 = -2.8297, A2 = 2.7745, A3 = 0.0552). The weight-change scenario was also computed with the same engine.
2. Tourism: A hotel chain's choice of new-concept restaurant operator
A hotel chain will choose among three prospective operators for a new-concept restaurant at its flagship hotel. The criteria are: originality of the culinary concept, the operating team's experience, and the risk of a rent/turnover-share dispute over the contract term; the last is a lower-is-better criterion. Because the hotel management has not worked with these candidates before, it has scored them using words chosen from the term set.
The method converts the three candidates' terms, together with their translation values, into numbers, builds the negative-ideal according to each criterion's direction, computes the two distances, and produces the assessment score through pairwise comparisons. Suppose the candidate with the most original concept is also the one with the highest dispute risk. It still comes out first, because the weight on originality exceeds the weight on risk. The second and third candidates' Euclidean distances come out close to one another, and the order between them is decided by the Taxicab distance.
Management's hesitation: choosing the most original concept also brings the risk of working with a less experienced team. It should also be stated in the report that the order between the second and third candidates comes from the Taxicab distance, and that these two could swap places if the threshold value were changed.
In the report: "With the high weight given to concept originality, the most original candidate reaches the highest assessment score; the order between the second and third candidates is sensitive to the threshold value, and this candidate's team experience should be monitored as a separate risk."
3. What Not to Do
The first error, in the illustrative example, is discarding the translation value on A3's interface-usability term and rounding it straight to "fairly high." The ranking does not change, but the information about how robust the gap between two alternatives is gets lost. The second error is additionally dividing the term-to-number converted values by the column maximum; this needlessly adds crisp CODAS's normalisation and shifts the negative-ideal point. The third error is leaving the threshold value (ψ) at its default without ever questioning it; even though the large gap between A2 and A3 in this example makes it unimportant here, this test must not be skipped in another table where the alternatives come out close.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/l2t-codas
Keshavarz Ghorabaee, M., Zavadskas, E. K., Turskis, Z., & Antucheviciene, J. (2016). A new combinative distance-based assessment (CODAS) method for multi-criteria decision-making. Economic Computation and Economic Cybernetics Studies and Research, 50(3), 25–44. (no DOI. This article is not registered in Crossref; see the Sources section of the base CODAS card.)
Herrera, F., & Martínez, L. (2000). A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Transactions on Fuzzy Systems, 8(6), 746–752. DOI: 10.1109/91.890332
Zadeh, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning—I. Information Sciences, 8(3), 199–249. DOI: 10.1016/0020-0255(75)90036-5
Aires, R. F. de F., & Ferreira, L. (2018). The rank reversal problem in multi-criteria decision making: A literature review. Pesquisa Operacional, 38(2), 331–362. DOI: 10.1590/0101-7438.2018.038.02.0331